Non-positivity of the semigroup generated by the Dirichlet-to-Neumann operator

Abstract

By analysing some explicit examples we investigate the positivity and non-positivity of the semigroup generated by the Dirichlet-to-Neumann operator associated with the operator $\mathrm{\Delta }+\lambda I$ as $\lambda$ varies. It is known that the semigroup is positive if $\lambda <{\lambda }_1$, where ${\lambda }_1$ is the principal eigenvalue of $-\mathrm{\Delta }$ with Dirichlet boundary conditions. We show that it is possible for the semigroup to be non-positive, eventually positive or positive and irreducible depending on $\lambda >{\lambda }_1$.